Flutter Stability and Aerodynamic Optimization of Cable-Stayed Bridge Deck Using Numerical Simulation

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1 Flutter Stability and Aerodynamic Optimization of Cable-Stayed Bridge Deck Using Numerical Simulation Saad A.Yehia, Walid A.Attia Abstract The importance of bridge aerodynamic investigations was immediately realized after the Tacoma Narrows Bridge collapsed in Since, the cable-stayed bridges are prone to the aerodynamic instabilities caused by wind this becomes a prime criterion to be checked during the design. If the wind velocity exceeds the critical velocity for flutter that the bridge can withstand, then the bridge fails due to the phenomenon of flutter. This thesis consists of FEM modal analysis of the Tatara Bridge via FEM software ANYSYS. In the bridge modal analysis, the lowest three lateral, vertical and torsional modes and their corresponding frequencies are calculated through Lanczos method solver in ANSYS. Therefore, the target is to optimize the deck shape to reduce the aerodynamic forces. To achieve this goal, more than 30 model cases were tested in order to obtain an optimized aerodynamic configuration of the deck. The influence of the geometry modifications on the aerodynamic stability has been established through this thesis, which has beneficial to the final design of the bridge. With a strong computational background, CFD (Computational Fluid Dynamics) simulations appear to be powerful rivals of the wind tunnel tests, which are expensive, require a scale model and a time consuming tool in designing bridges. Therefore, the analyses are carried out for deck shapes through CFD software OpenFOAM V2.3.1, establishing a dimensional fluid-structure interaction (FSI) numerical model to calculate the flutter critical wind speed. The result of this models shows the optimized deck shapes can significantly reduce the aerodynamic forces. Index Terms Tatara Bridge, Computational fluid dynamics, Critical flutter wind speed, Fluid-Structure Interaction. I. INTRODUCTION Wind load is one of the most important design loads in civil engineering structures, especially for long span bridges with low damping and high flexibility. Deck sections of long span bridges are one type of bluff bodies that are usually elongated with sharp corners that make the flow around them to cause aerodynamic instabilities. Such instabilities may cause serious catastrophic structural failure such as, the Old Tacoma Narrows Bridge collapse in Cable stayed bridges not only must be designed to support static wind forces like lift, drag and moment created by the mean wind, but also the dynamic loads created by an interaction between the wind forces and structural motions. There are many cable-stayed bridges that could be chosen as a case studies. The case study presented here is the Tatara cable-stayed bridge. The Tatara Bridge is located on the Western Expressway along with eight other bridges and was completed in The Western Expressway goes through nine of the Geiyo Islands with the Tatara Bridge connecting two of these archipelagos together. The Honshū-Shikoku Bridge Authority was set up to connect the largest island in Japan, Honshū with the smallest, and Shikoku which was previously only reached by ferry. The project consists of three expressways Central, Eastern and Western. Fig.1 shows the perspective of Tatara Bridge. Tatara Bridge is located in one of the most geologically active parts of the world and designed for some of the world s biggest typhoons prevalent in Japan. In addition, it is located in a very geologically active area and has to Fig.1 Perspective of Tatara Bridge 209

2 withstand major earthquakes. The original proposal for the Tatara Bridge was to be a suspension bridge, however, due to the effect it would have on the surrounding national park area it was changed to a cable stayed bridge. With the main span of 890m at the time of construction, it was the longest cable-stayed bridge in the world. Tatara Bridge confirmed that a 900m span bridge has the load-bearing capacity needed especially under extreme wind loading. Fig. 2 shows the dimensions of the Bridge. Fig.2 Dimensions of the Bridge [1] The main tower is 220 m high and designed as an inverted Y shape. It has a cross-shaped section with corners cut for higher wind stability and better landscaping. (Material properties, G= 8.10E+09 kg/m2, E= 2.10E+10 kg/m2, TC= 1.20E-05). The main girder section, as shown in Fig.3, consists of three spans, 270 m, 890 m, and 320 m, and measures 1480 m in total length. As either side span is shorter than the center span, PC girders are installed at each end of both side span sections as counterweight girders to resist negative reaction. This cable stayed bridge thus uses a steel and PC connection girder. The bridge has a total width of 30.6 m, including a road for motorized bicycles and pedestrians (hereafter called sidewalk) and a girder height of 2.7 m. It uses flat box girders attached with fairings to ensure wind stability. (Prestressed concrete sections properties, G= 1.22E+09 kg/m2, E= 2.80E+06 t/m2, TC= 1.00E-05 and steel sections properties, G=8.10E+09 kg/m2, E= 2.10E+10 kg/m2, TC = 1.20E-05). A. Finite Element Structural Modeling Fig.3 General Arrangement (main girder) II. METHODOLOGY A spine-type tridimensional nonlinear FE model of the Tatara Bridge is developed in the ANSYS environment. A graphical view of the model is presented in Fig.4. The orthotropic deck is modeled by an equivalent longitudinal frames. The frames of the deck are reproduced by means of BEAM4 elements, i.e. tridimensional elastic beams, capable for large displacements and small strains. A null mass density and mass weight is assigned to these elements, while lumped equivalent masses (MASS 21 elements), with rotary inertia in the longitudinal direction, are placed along the deck, in correspondence of each cable. Towers consist of both equivalent and variable sections, so they are reproduced by both BEAM4 and BEAM44 elements. Modeling of the stay cables is possible in ANSYS by employing the 3-D tension-only truss elements (LINK10), and utilizing its stress-stiffening capability. With this element, the stiffness is removed if the element goes into compression, thus simulating a slack 210

3 cable. No bending stiffness is included, whereas the pre-tensions of the cables can be incorporated by the initial strains of the element. Each stay cable is modeled by one element, which results in 168 tension-only truss elements in the model. Fig.4 3D Finite element model of the Tatara Bridge The modal analysis needs to solve the eigenvalue problem. The eigenvalue and eigenvector extraction technique used in the analysis is the Block Lanczos method. The Block Lanczos eigenvalue extraction method is available for large symmetric eigenvalue problems. Typically, this solver is applicable to the type of problems solved using the Subspace eigenvalue method, however, at a faster convergence rate. The Block Lanczos algorithm is basically a variation of the classic Lanczos algorithm, where the Lanczos recursions are performed using a block of vectors as opposed to a single vector. B. Computational Fluid Dynamics (CFD) In recent years, with rapid development of computer technique, some universal CFD software, such as OpenFOAM, Fluent, CFX, and so on, were adopted by bridge design organization because of good interface, convenient pre-processor and post-processor, open secondary developing function and so on. Therefore, the analyses are carried out for deck shapes by numerical simulations. Flutter occurs due to a structure and wind interaction where the wind speed has passed the critical speed of flutter and negative damping develops [2]. If a structure is experiencing oscillation a positive damping will slowly decrease the amplitude of displacement, on the other hand flutter increases the amplitude of the oscillation as time continues [3]. Fig.5 shows a sinusoidal representation of both positive and negative damping phenomena. (a) Positive damping (b) Negative damping Fig.5 Example of positive and negative damping [4] C. Numerical Simulation Principle The structure is regarded as a mass, spring and damping system. A schematic diagram of numerical simulation is shown in Fig.6. Fluid control equations for incompressible flow are given in (1), (2) which represent the continuity 211

4 and the Navier-Stokes equation respectively. The first step to ascertain the aerodynamic response of the considered bridge deck types is computation of the aerodynamic force coefficients (C d, C l, and C m ). After getting these F coefficients, forces ( D, F L, and M) can be easily calculated using (3), (4), and (5) [5]. Fig.7 shows criteria for the aerodynamic forces and moment. Equations (6), (7) are the governing structural equations for the heaving and torsional mode [2].. V 0 (1) V 1 2 ( V. ) V p V t p (2) F D F L U BCd (3) U BC l (4) 2 M 0.5 U BC m (5) mh( t ) C h( t ) K h( t ) F ( t ) (6) h h L I ( t ) C ( t ) K ( t ) M ( t ) (7) Fig. 6 Schematic diagram of numerical simulation Fig. 7 Sign criteria for the aerodynamic Forces [5]. Where: V, p and t: Velocity, pressure, time respectively. : Air density. : Air dynamic viscosity F D F, L, and M: Drag force, lift force, and moment respectively. C d, C l, and C m : Coefficients of drag force, lift force, and moment respectively. U: Reference velocity. B: Bridge width. m: Deck mass per unit length. I: Mass moment of inertia about shear center per unit length. C, Cα: Structural damping coefficients. h K, Kα: Translational and rotational spring stiffness. h h( t ), h( t ), h( t ) : Instantaneous bending acceleration, velocity and displacement respectively. ( t ), ( t ), ( t ) : Instantaneous torsional acceleration, velocity and displacement respectively. The procedure of FSI simulation in every wind speed is shown in Fig

5 Fig. 8 Procedure of FSI in every wind speed [6] Before calculating the time step, the preliminary value of bending and torsional acceleration, velocity, and displacement are set to be zero [6], [7]. For every time step the pressure and velocity are computed around the bridge deck for the given position by solving the continuity and Navier-Stokes equations as in (1), (2). Then the aerodynamic force coefficients acting on the bridge deck are calculated by using (4), (5). Lift pressure force and moment are represented by the force in y-direction and the force that causes rotation respectively. Lift force is applied at the center of gravity and the moment is applied at the shear center, then the lift and moment are extracted into structural dynamic equations (6), (7). Then they are solved by using The Newmark-β method to get the displacements for the heave and pitch. These displacements are applied in a rigid body fashion and the grid is updated. The velocity of the grid is applied from one time step to the next one by dividing the time step size in different positions. This process is repeated for several time steps. Then the velocity of the grid is extracted in the Navier stokes equation to account and simulate deck movement by a dynamic mesh technique. D. Numerical Simulation Model The bridge deck section was studied numerically using a CFD software in order to create an empirical reference set for numerical investigations. Table 1 shows all full scale parameters for it. The open source code OpenFOAM V2.3.1, based on the Finite Volume Method, is used to numerically evaluate the flow field. The turbulent flow around the mentioned bridge deck is modelled by the RANS with -ω-sst approach. The shear stress transport (SST) -ω models the Reynolds stresses with two transport equations for the turbulent kinetic energy and the specific dissipation rate ω. The algorithm used to solve the Reynold s Averaged Navier-Stokes equations is PIMPLE, an incompressible transient turbulent flow solver, which combines the PISO and SIMPLE algorithms for computing the pressure. The PIMPLE algorithm is compiled in the OpenFOAM solver, pimpledymfoam, and was 213

6 used in all the computations presented herein. PISO stands for Pressure Implicit with Splitting the Operators algorithm while SIMPLE represents Semi-Implicit Method for Pressure-Linked Equation algorithm [8]. Table 1. Full scale properties of the deck section Parameters Units Values f Natural vertical frequency ( v ) (see Table 3) Hz f Natural torsional frequency ( t ) (see Table 3) Hz Mass per unit length (m) Kg/m Mass moment of inertia about shear center per unit length (I) Kg.m² /m *10 The computational region and boundary conditions of the bridge deck are shown on Fig.9. As shown in Fig.9, the height of the fluid domain is 8B and the length is 16B where (B) is the deck width. Fig.9 Computational region and boundary conditions of the bridge deck For all the simulation presented in this work, the pressure is enforced as zero gradient at the inlet of the tunnel and zero value at the outlet, while the velocity is fixed at the inlet and has a zero gradient boundary condition at outlet. The upper and lower sides are specified as symmetrical. The no-slip boundary condition is applied on the deck surface. The OpenFOAM boundary condition settings for velocity and pressure are given in Table 2. Table 2. Boundary conditions for velocity and pressure Boundary BC for velocity BC for pressure Inlet Fixed value Zero Gradient Outlet Zero Gradient Fixed value (0) Top and Bottom Zero Gradient Zero Gradient Deck Moving wall velocity (0) Zero Gradient For the meshing of deck geometry generated with SOLIDWORKS, the OpenFOAM utilities blockmesh, surfacefeatureextract, and snappyhexmesh are used. These utilities allow the user to define the domain and break it up into a coarse mesh (blockmesh), then define feature edges in the geometry that should have sharp edges in the final mesh (surfacefeatureextract), and, finally, create the refined mesh using snappyhexmesh which snaps the coarse mesh to the surface of the geometry. The snappyhexmesh application also permits the user to define certain mesh quality criteria (e.g. maximum skewness of cells, maximum non-orthogonality of cells) to control the final mesh. Fig.10 shows a section of the mesh created by the snappyhexmesh OpenFOAM application. The final mesh consisted of cells and Nodes. 214

7 Fig.10 Mesh of the computation region of the bridge deck After choosing the solver and discretization schemes, the simulation is run with OpenFOAM. Postprocessing software ParaView is used to visualize the results of the computation for the user and to calculate key figures. A. Characteristics of the natural mode shapes III. SIMULATION RESULTS Cable-stayed bridges are more flexible than other structures because of large spans. One important aspect of such a flexible structure is a large displacement response of the deck when subject to dynamical loads. As a result, considerable amount of work has been conducted to study the dynamic behavior of cable-stayed bridges as a part of the design of wind and seismic resistance. The dynamic characteristics of a structure can be effectively analyzed in terms of natural frequencies and mode shapes of the cable-stayed bridges. The natural frequencies and mode shapes of the Tatara Bridge are studied by using the current finite element model. Since the established modal is a 3-D finite element model, a general model analysis is capable to provide all possible modes of the bridge (transverse, vertical, torsion, and coupled). The lowest three lateral, vertical and torsional modes and their corresponding frequencies of the bridge shown in Table 3 and Fig.11. Table 3. Characteristics of the lowest natural mode shapes Modes Eigenvalue(ω²) Frequency (Hz) Mode shape E-02 Anti-symmetric Lateral modes E-02 Symmetric E-02 Anti-symmetric Symmetric Vertical modes Symmetric Anti-symmetric Symmetric Torsional modes Symmetric Symmetric 215

8 Lateral modes ISSN: f=2.41e-02 HZ f=3.84e-02 HZ f=5.528e-02 HZ Vertical modes f= HZ f= HZ f= HZ Torsional modes f= HZ f= HZ f= HZ Fig.11 Natural frequencies and mode shapes for the lowest three lateral, vertical and torsional modes for the Tatara Bridge 216

9 B. The Influence of Section Rostra on Flutter Critical Wind Speed Because the critical wind speed is sensitive to shape of section rostra [9], [10], [11], rostra with different width and acutance is taken into account in the tests. The acutance varies from 55 deg to 29 deg, correspondingly the width varying from 0.75m to 2.50m. Total eight model cases were tested. Flutter wind speeds were obtained in the tests for the section model with attack angles (-3 deg, 0 deg and +3 deg) and the results are shown in Table 4 and Fig.12. It is noted that the flutter critical wind speed decreases along with the increasing of rostra width and acutance. Table 4. Critical wind speed varying with different section rostra case Type of the section rostra Flutter critical wind speed (m/s) Width : 0.75m Width : 1.0m Width : 1.25m Width :1.50m Width :1.75m Width : 2.00m Width : 2.25m Width : 2.50m

10 Fig.12 Flutter wind speed varying with the acutance of rostra To find the critical wind speed of flutter for each case, time history analysis for aerodynamic coefficients and vibrating motion should be applied by increasing the inlet velocity incrementally in different runs. When the aerodynamic coefficients and motion amplitude started to grow (negative damping), the critical velocity was found. From Fig.13 for case (1) it can be seen that: When wind speed equals m/sec, lift coefficient decrease with the increase of time. This illustrates that the total damping of the model is positive. When wind speed equals m/s, lift coefficient remain almost the same. When wind speed reaches m/sec, lift coefficient increase with the increase of time. This illustrates that the total damping of the model changes from positive to negative. So flutter critical wind speed equals m/sec. Cl (a)v= m/s Cl -0.2 Flow Time (sec) Flow Time (sec) (b)v= m/s 218

11 Cl ISSN: Flow Time (sec) (c)v= m/s Fig.13 Time histories of lift coefficient. C. The Influence of Lower Web Slope on Flutter Critical Wind Speed The rostra with varying of lower web slopes and the width of section rostra is only 0.50m is taken into account in the tests. The web slope varies from 5 deg to 20 deg. Total four model cases were tested. Flutter wind speeds were obtained in the tests for the section model and the results are shown in Table 5. It is noted that the flutter critical wind speed increases along with the increasing of the steepness of lower web slope. Table 5. Critical wind speed (width of section rostra is 0.50m) Case Section rostra Flutter critical wind speed (m/s) D. The Influence of Rostra Width with Fixed Steepness of Lower Inclined Web Slope on Flutter Critical Wind Speed The wider and acuminate section rostra are more difficult to be fabricated and fixed, implying more cost in design and construction, although it can strengthen the aerodynamic stability of the girder distinctly. Alternate way is to fixed steepness of lower inclined web slope and varying the rostra width. Flutter wind speeds were obtained in the tests for the section model and the results are shown in Table 6. It is noted that the flutter critical wind speed increases along with the increasing of the rostra width. 219

12 case ISSN: Table 6. Critical wind speed (web slope is 28 ) Type of the section rostra Flutter critical wind speed (m/s) Width : 0.0m Width : 0.15m Width :0.23m Width :0.33m Width :0.47m Width : 0.56m Width : 0.68m Width : 0.85m Width : 1.00m E. The Influence of the Curvature Section Rostra on Flutter Critical Wind Speed The rostra with varying of lower web slopes and the curvature section rostra were taken into account in the tests. The web slope varies from 28 deg to 20 deg. Total nine model cases were tested. Flutter wind speeds were obtained in the tests for the section model and the results are shown in Table 7. It is noted that the flutter critical wind speed increases along with the increasing of the curvature rostra 220

13 Table 7. Critical wind speed varying with different section rostra case Type of the section rostra Flutter critical wind speed (m/s) Radius :0.30m Radius :0.35m Radius :0.40m Radius :0.45m Radius :0.50m Radius :0.55m Radius :0.60m Radius :0.65m Radius :0.70m F. The Influence of Guide Wing on Flutter Critical Wind Speed The guide wing on the edge of sideway can smooth airflow while passing through the section. Hence the aerodynamic stability may be strengthened [10], [11], [12]. Total six different types of guide wings were applied in the tests, with different width and obliquity. Flutter wind speeds were obtained in the tests for the section model with attack angle 0 deg, and the results are shown in Table 8. It is observed that the wider and positive obliquity guide wing can improve the aerodynamic stability. On the other hand, the guide wing will increase the complexity of the structure design and construction, particularly in the location of the rostra, and the maintenance cost will 221

14 increase correspondingly. The guide wing of the deck is not recommended in the design unless there is no alternative means to improve the aerodynamic performance. The outline of the used guide wing shown in Fig.14. Case Table 8. Critical Wind Speed Varying With Different Guide Wings Guide wing Flutter critical wind speed (m/s) Width Obliquity Attack angle = m m m m m m G. Numerical Validations Fig.14 Outline of guide wing The first vertical and torsional modes of Tatara Bridge are compared with results obtained from other researcher as shown in Table 9. The results of the frequencies values are relatively accurate comparing to the numerical values. References Table 9. References of Frequencies for the Tatara Bridge Frequency(Hz) 1 st Vertical Mode 1 st Torsional Mode Present Work M.Ito et al.[13] The result of work done for the basic section of Tatara Bridge is compared with the Selberg formula [14] as shown in Table 10. The critical flutter velocity predicted in the present work is a good agreement with the Selberg formula. Table 10. References of Flutter Velocity for the Basic Section of Tatara Bridge References Vcr (m/s) Flutter Error (ΔVcr %) Present Work (Basic Section) 124 Selberg Formula 116 IV. CONCLUSIONS The following points offer the major outcome of the present study: 1. The study of the Tatara Bridge has shown that automatic updating of FE model using ANSYS software is feasible. It can thus improve prediction of the behaviour of bridge structure and it could be used as a tool in a bridge assessment process. 2. The aerodynamics of bridge deck cross section has been fully described through CFD simulations by using OpenFOAM software. 3. FSI is considered as a direct simulation method for the flutter stability of bridge and was developed based on CFD OpenFOAM software and proved to be useful in the early aerodynamic design stage of cable stayed bridges. 4. Using a guide wing can smooth airflow while passing through the section and the aerodynamic stability will be strengthened. On the other hand, the guide wing will increase the complexity of the structure design,

15 construction and the maintenance cost. The guide wing of the deck is not recommended in the design unless there is no alternative means to improve the aerodynamic performance. 5. The wider and acutance section rostra can strength the aerodynamic stability of the girder. 6. The flutter critical wind speed is sensitive to the steepness of blow inclined web slope. The lower is the slope, the higher is wind speed. 7. When the slope of lower inclined web is 28 deg, the flutter critical Wind speed increases along with the increasing of the rostra width. 8. Using the curvature section rostra will increase the flutter critical wind speed with in the increasing of the curvature rostra. On the other hand, the curvature section rostra will increase the complexity of the structure design and construction cost. 9. The results also lead to an optimized section of girder: shorter rostra and without guide wing. It also satisfies different kinds of requirements: high security, low cost, and more convenience. 10. Through the present work, the final section of the Tatara Bridge has a good aerodynamic stability based on the flutter critical wind speed. ACKNOWLEDGMENT The author Saad A. Yehia thanks Eng. Mohamed Abou-Elela for the many valuable comments to the manuscript. REFERENCES [1] Yabuno, M.Fujiwara, T.Sumi, and K.Nose, Design of Tatara Bridge, Engineering Review, vol. 36, no. 2, pp.40-56, June [2] Hao Zhan, Tao Fang, and Zhiguo Zhang, Flutter stability studies of Great Belt East suspension bridge by two CFD numerical simulation methods, 6 th European & African Conference on Wind Engineering, Robinson College, Cambridge, UK, 7-11July [3] W.M.Zhang, Y.J.Ge, and M.L.Levitan, Aerodynamic flutter analysis of a new suspension bridge with double main spans, journal of wind and structure, 14(3), [4] Vikas Arora, Hybrid Viscous-Structural Damping Identification Method, Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science, vol.23, pp , [5] Fẽlix Nieto, Ibuki Kusano, Santigo Hernảndez, and Josẽ Ả. Jurado, CFD analysis of the vortex shedding response of a twin box deck cable stayed bridge, The Fifth International Symposium on Computational Wind Engineering Chapel Hill, North Carolina, USA, May [6] Xiaobing Liu, Zhengqing, and Chenb Zhiwen Liu, Direct simulation method for flutter stability of bridge deck, The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7), China, 2-6 September [7] Gergely Szabό, Jόzsef Gyὅrgyi, and Gergely Kristόf, Advanced flutter simulation of flexible bridge decks, Coupled Systems Mechanics, vol. 1, no. 2, pp , [8] Jasak H, Error analysis and estimation for the finite volume method with applications to fluid flows, PhD thesis, Imperial College of Science, Technology and Medicine, [9] Allan Larsen, Aerodynamic aspects of the final design of the 1624 m suspension bridge across the Great Belt, Journal of Wind Engineering and Industrial Aerodynamics, vol. 48, Issues 2-3, pp , [10] Toshio Miyata, Historical view of long-span bridge aerodynamics, Journal of Wind Engineering Industrial Aerodynamic (91), pp , [11] Song Jinzhong, Lin Zhixing, Xu Jianying, Research and Appliance of Aerodynamic Measures about Wind resistance of Bridges, Journal of Tongji University, vol. 30, no. 5, May [12] K. Wilde, P. Omenzetter, Y. Fujino, Suppression of bridge flutter by active deck-flaps control system, 127(1), pp ,

16 [13] M. Ito, T. Endo, T. Iijima, and A. Okukawa, The technical challenge of a long cable-stayed bridge- Tatara Bridge, Department of Construction Engineering, Saitama University, Japan, [14] Selberg A. Oscillation and aerodynamic stability of suspension bridges, Acta. Polytech. Scand., 13, pp , AUTHOR BIOGRAPHY First Author: PhD Student in Structure Dept., Faculty of Eng., Cairo University, Giza, Egypt. Eng_saadyhy@yahoo.com Second Author: Professor of Theory of Structure, Structure Dept., Faculty of Eng., Cairo University, Egypt. Waattia@gmail.com 224